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 A027640 PoincarĂ© series [or Poincare series] for ring of modular forms of genus 2. 3
 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 2, 0, 4, 0, 4, 0, 5, 0, 6, 0, 8, 0, 7, 0, 10, 0, 11, 0, 12, 0, 14, 1, 17, 0, 16, 1, 21, 1, 22, 1, 24, 2, 27, 3, 31, 2, 31, 4, 37, 4, 39, 5, 42, 6, 46, 8, 52, 7, 52, 10, 60, 11, 63, 12, 67, 14 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,11 COMMENTS a(k) for k>0 is the dimension of the space of Siegel modular forms of genus 2 and weight k (for the full modular group Gamma_2). - Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009 REFERENCES B. Runge, On Siegel modular forms I, J. Reine Angew. Math., 436 (1993), 57-85. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 J. Igusa, On Siegel modular forms of genus 2 (II), Amer. J. Math., 86 (1964), 392-412, esp. p. 402. Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,1,1,0,0,-1,-1,-1,1,0,0,1,-1,-1,-1,0,0,1,1,1,0,0,0,-1). FORMULA G.f.: (1+x^35)/((1-x^4)*(1-x^6)*(1-x^10)*(1-x^12)). MATHEMATICA Table[SeriesCoefficient[Series[(1+t^(35))/((1-t^4) (1-t^6)(1-t^(10)) (1-t^(12))), {t, 0, 100}], i], {i, 0, 100}] (* Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009 *) PROG (PARI) Vec((1 - x + x^2 - x^3 + x^4 - x^5 + x^6)*(1 + x - x^5 - x^6 - x^7 - x^8 + x^10 + x^11 + x^12 + x^13 + x^14 - x^16 - x^17 - x^18 - x^19 + x^23 + x^24) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2*(1 - x^2 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^76)) \\ Colin Barker, Jul 27 2019 CROSSREFS Cf. A165685 for the corresponding dimension of the space of cusp forms. - Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009 Sequence in context: A128145 A128143 A292561 * A349448 A194666 A325799 Adjacent sequences:  A027637 A027638 A027639 * A027641 A027642 A027643 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified May 28 16:37 EDT 2022. Contains 354119 sequences. (Running on oeis4.)